5.36. Решите уравнения:

Решение:

1. \( |x| = 5 \)
   \( x = 5 \) или \( x = -5 \)
2. \( |x| = 0 \)
   \( x = 0 \)
3. \( |x| = -2 \)
   решений нет, так как модуль числа не может быть отрицательным.
4. \( |x| = 1,2 \)
   \( x = 1,2 \) или \( x = -1,2 \)
5. \( |x| = 12,009 \)
   \( x = 12,009 \) или \( x = -12,009 \)
6. \( |x-3| = 1 \)
   \( x-3 = 1 \) или \( x-3 = -1 \)
   \( x = 4 \) или \( x = 2 \)
7. \( |x+4| = 5 \)
   \( x+4 = 5 \) или \( x+4 = -5 \)
   \( x = 1 \) или \( x = -9 \)
8. \( |2x-1| = 1 \)
   \( 2x-1 = 1 \) или \( 2x-1 = -1 \)
   \( 2x = 2 \) или \( 2x = 0 \)
   \( x = 1 \) или \( x = 0 \)
9. \( |-1 \frac{23}{24}| \)
   \( |-\frac{47}{24}| = \frac{47}{24} \) — это число. Нет уравнения.
10. \( |5x-10| = 0 \)
    \( 5x-10 = 0 \)
    \( 5x = 10 \)
    \( x = 2 \)
11. \( |x+57| = 13 \)
    \( x+57 = 13 \) или \( x+57 = -13 \)
    \( x = 13-57 \) или \( x = -13-57 \)
    \( x = -44 \) или \( x = -70 \)
12. \( |2x-13| = 3 \)
    \( 2x-13 = 3 \) или \( 2x-13 = -3 \)
    \( 2x = 16 \) или \( 2x = 10 \)
    \( x = 8 \) или \( x = 5 \)
13. \( |4x-5| = 9 \)
    \( 4x-5 = 9 \) или \( 4x-5 = -9 \)
    \( 4x = 14 \) или \( 4x = -4 \)
    \( x = \frac{14}{4} = \frac{7}{2} \) или \( x = -1 \)
14. \( |3x-13| = -4 \)
    решений нет.
15. \( |12x-1| = 0 \)
    \( 12x-1 = 0 \)
    \( 12x = 1 \)
    \( x = \frac{1}{12} \)
16. \( |2x+7| = 11 \)
    \( 2x+7 = 11 \) или \( 2x+7 = -11 \)
    \( 2x = 4 \) или \( 2x = -18 \)
    \( x = 2 \) или \( x = -9 \)
17. \( |2x-9| = -1 \)
    решений нет.
18. \( |x+4| = 11 \)
    \( x+4 = 11 \) или \( x+4 = -11 \)
    \( x = 7 \) или \( x = -15 \)
19. \( |15x-7| = 8 \)
    \( 15x-7 = 8 \) или \( 15x-7 = -8 \)
    \( 15x = 15 \) или \( 15x = -1 \)
    \( x = 1 \) или \( x = -\frac{1}{15} \)
20. \( |4-5x| = \frac{1}{2} \)
    \( 4-5x = \frac{1}{2} \) или \( 4-5x = -\frac{1}{2} \)
    \( -5x = \frac{1}{2} - 4 = -\frac{7}{2} \) или \( -5x = -\frac{1}{2} - 4 = -\frac{9}{2} \)
    \( x = \frac{7}{10} \) или \( x = \frac{9}{10} \)
21. \( |0,3x-7| = \frac{2}{3} \)
    \( 0,3x-7 = \frac{2}{3} \) или \( 0,3x-7 = -\frac{2}{3} \)
    \( \frac{3}{10} x = 7 + \frac{2}{3} = \frac{23}{3} \) или \( \frac{3}{10} x = 7 - \frac{2}{3} = \frac{19}{3} \)
    \( x = \frac{23}{3} \cdot \frac{10}{3} = \frac{230}{9} \) или \( x = \frac{19}{3} \cdot \frac{10}{3} = \frac{190}{9} \)
22. \( |2x-1| = \frac{1}{7} \)
    \( 2x-1 = \frac{1}{7} \) или \( 2x-1 = -\frac{1}{7} \)
    \( 2x = 1 + \frac{1}{7} = \frac{8}{7} \) или \( 2x = 1 - \frac{1}{7} = \frac{6}{7} \)
    \( x = \frac{4}{7} \) или \( x = \frac{3}{7} \)
23. \( |0,2x-3,5| = 2,7 \)
    \( 0,2x-3,5 = 2,7 \) или \( 0,2x-3,5 = -2,7 \)
    \( 0,2x = 2,7+3,5 = 6,2 \) или \( 0,2x = -2,7+3,5 = 0,8 \)
    \( x = \frac{6,2}{0,2} = 31 \) или \( x = \frac{0,8}{0,2} = 4 \)
24. \( |0,7x-1| = 1 \)
    \( 0,7x-1 = 1 \) или \( 0,7x-1 = -1 \)
    \( 0,7x = 2 \) или \( 0,7x = 0 \)
    \( x = \frac{2}{0,7} = \frac{20}{7} \) или \( x = 0 \)
25. \( |7x-5| = \frac{2}{3} \)
    \( 7x-5 = \frac{2}{3} \) или \( 7x-5 = -\frac{2}{3} \)
    \( 7x = 5 + \frac{2}{3} = \frac{17}{3} \) или \( 7x = 5 - \frac{2}{3} = \frac{13}{3} \)
    \( x = \frac{17}{21} \) или \( x = \frac{13}{21} \)

Ответ: 1) x = ±5; 2) x = 0; 3) решений нет; 4) x = ±1,2; 5) x = ±12,009; 6) x = 4, x = 2; 7) x = 1, x = -9; 8) x = 1, x = 0; 9) решений нет; 10) x = 2; 11) x = -44, x = -70; 12) x = 8, x = 5; 13) x = 7/2, x = -1; 14) решений нет; 15) x = 1/12; 16) x = 2, x = -9; 17) решений нет; 18) x = 7, x = -15; 19) x = 1, x = -1/15; 20) x = 7/10, x = 9/10; 21) x = 230/9, x = 190/9; 22) x = 4/7, x = 3/7; 23) x = 31, x = 4; 24) x = 20/7, x = 0; 25) x = 17/21, x = 13/21.